MONDAY TUESDAY WEDNESDAY THURSDAY FRIDAY
  9:00-9:45 Bogdan 9:00-9:45 Dolbeault 9:00-9:45 Garofalo 9:00-9:45 Zhang (online)
10:00-10:45 Golse 9:50-10:35 Rhandi 9:50-10:35 Kogoj 9:50-10:35 Tralli 9:50-10:35 Priola
10:55-11:40 Silvestre CoffeeBreak CoffeeBreak CoffeeBreak CoffeeBreak
CoffeeBreak 11:10-11:55 Bramanti 11:10-11:55 Nyström 11:10-11:55 Pallara 11:00-11:25 Marino
12:10-12:55 Imbert 12:00-12:45 Stroffolini 12:00-12:45 Lindgren 12:00-12:45 Metafune 11:30-11:55 Lucertini
Lunch Lunch Lunch Lunch 12:00-12:25 Pesce
14:30-15:15 Kohatsu Higa (online) 14:15-14:40 Rebucci   14:15-14:40 Lenczewska 12:30-13:15 Pagliarani
15:20-16:05 Lanconelli 14:45-15:10 Anceschi   14:45-15:10 Rutkowski Lunch
CoffeeBreak 15:15-15:40 Piccinini   15:15-15:40 Szczypkowski  
16:30-17:15 Jabir CoffeeBreak   CoffeeBreak  
17:20-18:05 Bally (online) 16:00-16:25 Kaleta   16:00-16:25 Litsgård  
  16:30-16:55 Sztonyk   16:30-16:55 Grzywny  
  17:00-17:25 Klimsiak   17:00-17:25 Spina  

Monday, June 13th

10:00-10:45

F. Golse (Ecole Polytecnique)

Title: Partial Regularity in Time for the Landau Equation with Coulomb Interaction

Abstract: In 1998, Villani proved the global existence of a class of weak solutions of the Cauchy problem for the spatially homogeneous Landau equation with Coulomb interaction for all initial data of finite mass, energy and entropy. Such weak solutions are referred to as “H solutions” since their definition involves the entropy production rate. To this date, it is unknown whether H solutions with smooth and rapidly decaying initial data remain smooth for all times or blow up in finite time. The purpose of this talk is to show that the set of times at which singularities may occur is “small”: specifically, we prove that its Hausdorff dimension is at most $1/2$.
(Work in collaboration with M.P. Gualdani, C. Imbert and A. Vasseur).

10:55-11:40L. Silvestre (Chicago University)

Title: Hölder continuity up to the boundary for kinetic equations

Abstract: We consider a kinetic Fokker-Planck equation with rough coefficients and the spatial variable restricted to a bounded domain. We study the Hölder continuity of the solutions on the boundary.


CoffeeBreak

12:10-12:55C. Imbert (CNRS, Paris)

Title: Partial regularity for the Landau equation with very soft potentials

Abstract: We will consider the Landau equation with very soft potentials, including Coulomb. We will see how to localize the truncated entropy estimates obtained in a previous work with Golse, Gualdani and Vasseur and how to derive partial regularity in time and velocity. We will also discuss the radial case for Coulomb potentials. Joint work with Golse and Vasseur.


Lunch

14:30-15:15A. Kohatsu Higa (Ritsumeikan University)
online talk (Google Meet link)

Title: Joint density of the stable process and its supremum: regularity and upper bounds

Abstract: We present a technique which uses a combination of three ideas from simulation to establish a nearly optimal polynomial upper bound for the joint density of the stable process and its associated supremum at a fixed time on the entire support of the joint law. The representation of the concave majorant of the stable process and the Chambers-Mallows-Stuck representation for stable laws are used to define an approximation of the random vector of interest.  An interpolation technique using multilevel Monte Carlo is applied to accelerate the approximation, allowing us to establish the infinite differentiability of the joint density as well as nearly optimal polynomial upper bounds for the derivatives of any order.
This is joint work with Jorge Gonzalez-Cazares and Alex Mijatovic ( University of Warwick)

15:20-16:05A. Lanconelli (Università di Bologna)

Title: A small time approximation for the solution to the Zakai Equation

Abstract: We propose a novel small time approximation for the solution to the Zakai equation from nonlinear filtering theory. We prove that the unnormalized filtering density is well described over short time intervals by the solution of a deterministic partial differential equation of Kolmogorov type; the observation process appears in a pathwise manner through the degenerate component of the Kolmogorov’s type operator. The rate of convergence of the approximation is of order one in the lenght of the interval. Our approach combines ideas from Wong-Zakai-type results and Wiener chaos approximations for the solution to the Zakai equation. The proof of our main theorem relies on the well-known Feynman-Kac representation for the unnormalized filtering density and careful estimates which lead to completely explicit bounds. Joint work with Ramiro Scorolli.


CoffeeBreak

16:30-17:15J-F. Jabir (HSE, Moscow)

Title: Lagrangian stochastic models for turbulent flows and related problems

Abstract: Lagrangian stochastic models define a particular class of Langevin type dynamics which originally emerged for the statistical description and simulation of turbulent flows. These models carry various singularities ranging from highly irregular nonlinear (in the sense of McKean) coefficients, to path or distributional constraints when physical boundaries or incompressibility need to be modeled.  The talk will present different mathematical results – mostly related to well posedness problems – and some practical applications achieved over recent years.

17:20-18:05V. Bally (Université de Marne-la-Vallée)
online talk (Google Meet link)

Title: Construction of Boltzmann and McKean Vlasov type flows (the sewing lemma approach)

Abstract: We are concerned with a mixture of Boltzmann and McKean-Vlasov type equations, this means (in probabilistic terms) equations with coefficients depending on the law of the solution itself, and driven by a Poisson point measure with the intensity depending also on the law of the solution. Both the analytical Boltzmann equation and the probabilistic interpretation initiated by Tanaka have intensively been discussed in the literature for specific models related to the behavior of gas molecules. In this paper, we consider general abstract coefficients that may include mean field effects and then we discuss the link with specific models as well. In contrast with the usual approach in which integral equations are used in order to state the problem, we employ here a new formulation of the problem in terms of flows of endomorphisms on the space of probability measure endowed with the Wasserstein distance. This point of view already appeared in the framework of rough differential equations. Our results concern existence and uniqueness of the solution, in the formulation of flows, but we also prove that the ”flow solution” is a solution of the classical integral weak equation and admits a probabilistic interpretation. Moreover, we obtain stability results and regularity with respect to the time for such solutions. Finally we prove the convergence of empirical measures based on particle systems to the solution of our problem, and we obtain the rate of convergence. We discuss as examples the homogeneous and the inhomogeneous Boltzmann (Enskog) equation with hard potentials.
This is a joint work with A. Alfonsi.

Tuesday, June 14th

9:00-9:45K. Bogdan (Wroclaw University)

Title: Self-similar solution for Hardy operator

Abstract: We will discuss the large-time asymptotics of solutions to the heat equation for the fractional Laplacian with added subcritical or even critical Hardy-type potential. The asymptotics is governed by a self-similar solution of the equation, obtained as a normalized limit at the origin of the kernel of the corresponding Feynman-Kac semigroup. Interestingly, an Ornstein-Uhlenbect semigroup turns out to be an important tool for the analysis.
The paper is a joint work with P. Kim (Seoul), T. Jakubowski, and D. Pilarczyk (Wrocław) and can be found on arXiv at https://arxiv.org/abs/2203.02039.

9:50-10:35A. Rhandi (Università di Salerno)

Title: Kolmogorov operators on noncompact metric graph

Abstract: In this talk we first prove the existence of a classical solution to a class of parabolic problems with unbounded coefficients on metric star graphs subject to Kirchhoff-type conditions. The result is applied to the Ornstein-Uhlenbeck and the harmonic oscillator operators on metric star graphs. We give an explicit formula for the associated Ornstein-Uhlenbeck semigroup and give the unique associated invariant measure. We show that this semigroup inherits the regularity properties of the classical Ornstein-Uhlenbeck semigroup on $\mathbb R$ and compute its spectrum.


CoffeeBreak

11:10-11:55M. Bramanti (Politecnico di Milano)

Title: Schauder estimates for degenerate Kolmogorov operators with coefficients measurable in time

Abstract: We consider degenerate Kolmogorov-Fokker-Planck operators $L$ with linear drift $Y$, modeled on the class of operators $a_{ij}u_{x_{i}x_{j}}+Yu$ studied by Lanconelli-Polidoro 1994 which are translation invariant and homogeneous of degree two. Our coefficients $a_{ij}$ are bounded and Hölder continuous in space (w.r.t. a distance induced by $L$ in $\mathbb{R}^{N}$) and only bounded measurable in time. We prove global “partial Schauder a priori estimates”, controlling the supremum in $t$ of the Hölder quotient in $x$ of the derivatives $u_{x_{i}x_{j}}$ and $Yu$ in terms of the analogous norm of $Lu$ and the supremum of $u$. We also prove that the derivatives $u_{x_{i}x_{j}}$ are locally Hölder continuous in space and time.
This is a joint work with S. Biagi.

12:00-12:45B. Stroffolini (Università “Federico II” di Napoli)

Title: Schauder type estimates for degenerate Kolmogorov equations with Dini continuous coefficients

Abstract: We study the regularity properties of the second order linear operator in $\mathbb{R}^{N+1}$:
$$
\mathscr{L} u := \sum_{j,k= 1}^{m} a_{jk}\partial_{x_j x_k}^2 u + \sum_{j,k= 1}^{N} b_{jk}x_k \partial_{x_j} u – \partial_t u,
$$
where $A = \left( a_{jk} \right)_{j,k= 1, \dots, m}, B= \left( b_{jk} \right)_{j,k= 1, \dots, N}$ are real valued matrices with constant coefficients, with $A$ symmetric and strictly positive. We prove that, if the operator $\mathscr{L}$ satisfies Hörmander’s hypoellipticity condition, and $f$ is a Dini continuous function, then the second order derivatives of the solution $u$ to the equation $\mathscr{L} u = f$ are Dini continuous functions as well. We also consider the case of Dini continuous coefficients $a_{jk}$’s. A key step in our proof is a Taylor formula for classical solutions to $\mathscr{L} u = f$ that we establish under minimal regularity assumptions on $u$.


Lunch

14:15-14:40A. Rebucci (Università di Modena e Reggio Emilia)

Title: On the weak regularity theory for solutions to degenerate Kolmogorov equations

Abstract: We here discuss the weak regularity of solutions to degenerate Kolmogorov equations as we studied it in [2]. More precisely, we prove a Harnack inequality and the Hölder continuity for weak solutions to equation $\mathscr{L} u=f$, under the assumptions of measurable coefficients, integrable lower order terms and nonzero source term.
Our proof relies on the combination of three fundamental ingredients – boundedness of weak solutions, weak Poincaré inequality and Log-transformation – in the same spirit of the recent paper [3] for the Fokker-Planck equation. In particular, we prove the boundedness of weak solutions to equation $\mathscr{L} u= f$ using the Moser’s iterative method. Thus, we manage to lower the integrability assumption on the lower order coefficients and to handle a non-vanishing source term $f$, extending to this more general case the Moser’s iterative scheme proposed in [4] and subsequently in [1].
Finally, we introduce a function space $\mathcal{W}$, which is the most natural framework for the study of the weak regularity theory for operator $\mathscr{L}$ and allows us to prove a weak Poincaré inequality.

References

[1] F.Anceschi, S. Polidoro, and M.A. Ragusa, Moser’s estimates for degenerate Kolmogorov equations with non-negative divergence lower order coefficients, Nonlinear Analysis}: 1-19, (2019).
[2] F. Anceschi and A. Rebucci, A note on the weak regularity theory for degenerate Kolmogorov equations, preprint arXiv:2107.04441, (2021).
[3] J. Guerand and C. Imbert, Log-transform and the weak Harnack inequality for kinetic Fokker-Planck equations, preprint arXiv: 2102.04105, (2021).
[4] A. Pascucci and S. Polidoro, The Moser’s iterative method for a class of ultraparabolic equations, Commun. Contemp. Math., 6(3):395-417, (2004).

14:45-15:10F. Anceschi (Univeristà Politecnica delle Marche)

Title: On the fundamental solution to the Kolmogorov equation & applications to financial market modeling

Abstract: In this talk, we discuss some recent results on the fundamental solution associated to a class of ultraparabolic equations in divergence form with measurable coefficients arising in the pricing problem for (Asian) options. In particular, we focus on the proof of its existence, alongside with Gaussian upper and lower bounds for it. These results are part of a joint project with Annalaura Rebucci.

15:15-15:40M. Piccinini (Università di Modena e Reggio Emilia)

Title: The Dirichlet problem for a family of totally degenerate differential operators

Abstract: In this talk we investigate the boundary value problem related to a family of totally degenerate, in the sense of Bony, differential operators
$$
\mathcal{L} := t^{2\vartheta} \sum_{i=1}^m \partial^{2}{x_i}+\sum{i,j=1}^N b_{ij}x_j\partial_{x_i}-\partial_t, \qquad \vartheta \in \mathbb{N}.
$$
In particular, in the framework of the Potential Theory, we present some results concerning existence and uniqueness for the Perron-Wiener-Brelot solution to the Dirichlet problem related to $\mathcal{L}$. Moreover, we also state a Wiener-type criterium and an exterior cone condition for boundary $\mathcal{L}$-regularity. Our results apply to a wide family of strongly degenerate operators that includes the following example $\mathcal{L} = t^2\Delta_x + \langle x, \nabla_y \rangle -\partial_t$, for $(x,y,t) \in \mathbb{R}^N \times \mathbb{R}^{N} \times \mathbb{R}$.


CoffeeBreak

16:00-16:25K. Kaleta (Wrocław University)

Title: On large-time properties of compact Schrödinger semigroups

Abstract: We will present recent results, obtained jointly with R. Schilling (Dresden), on the long-time behaviour of semigroups corresponding to non-local Schrödinger operators involving generators of symmetric Lévy process in Euclidean spaces and confining potentials. We establish sharp two-sided estimates of the corresponding heat kernels for large times and identify a new general regularity property, which we call progressive intrinsic ultracontractivity, to describe the large-time evolution of the corresponding Schrödinger semigroup. We discuss various examples and applications of these estimates, for instance we characterize the heat content and apply it to study the quasi-ergodic properties of semigroups. Our examples cover a wide range of processes, including isotropic stable and relativistic Levy processes. We have to assume only mild restrictions on the growth, resp. decay, of the potential and the jump intensity of the free process.

16:30-16:55P. Sztonyk (Wrocław University of Science and Technology)

Title: Heat kernels of non-local Schrödinger operators with Kato potentials

Abstract: We study heat kernels of Schrödinger operators whose kinetic terms are non-local operators built for sufficiently regular symmetric Lévy measures with radial decreasing profiles and potentials belong to Kato class. Our setting is fairly general and novel — it allows us to treat both heavy- and light-tailed Lévy measures in a joint framework.
We establish a certain relative-Kato bound for the corresponding semigroups and potentials. This enables us to apply a general perturbation technique to construct the heat kernels and give sharp estimates of them. Assuming that the Lévy measure and the potential satisfy a little stronger conditions, we additionally obtain the regularity of the heat kernels. Finally, we discuss the applications to the smoothing properties of the corresponding semigroups. Our results cover many important examples of non-local operators, including fractional and quasi-relativistic Schrödinger operators.

17:00-17:25T. Klimsiak (Institute of Mathematics, Polish Academy of Sciences)

Title: Strong maximum principle for local and non-local Schrödinger operators with singular potential

Abstract: We study Schrödinger operators on $L^2(E;m)$ of the form $−A+V$ with singular potentials $V$.
We address the question posed by H. Brezis about the structure of the set ${u=0}$ for non-negative supersolutions to $-Au+Vu = 0$.
The class of operators $A$ we study includes, in particular, symmetric Lévy type operators and symmetric diffusions in divergence form,
with strictly positive Green functions. The class of potentials $V$ consists of positive smooth measures, which contains, in particular,
Coulomb potentials and harmonic potentials, as well as generalized potentials, i.e. positive Borel measures concentrated on m-negligible sets.
We propose a probabilistic method based on Feynman-Kac formula for supersolutions and probabilistic potential theory.

Wednesday, June 15th

9:00-9:45J. Dolbeault (Université Paris Dauphine)

Title: Entropy methods and hypocoercivity for large times asymptotics

Abstract: $L^2$-hypocoercivity methods in kinetic equations rely on functional inequalities which determine decay or convergence rates measured by entropy methods in Fokker-Planck type equations. The lecture will be devoted to an overview of various recent results. 

9:50-10:35A. E. Kogoj (Università di Urbino)

Title: Liouville-type theorems for Kolmogorov and Ornstein–Uhlenbeck operators

Abstract: We collect Liouville-type properties that hold true for Kolmogorov operators with constant coefficients and for their time-stationary counterpart, the Ornstein–Uhlenbeck operators. In particular, we discuss uniqueness results for solutions and sub- solutions in $L^p$-spaces, for solutions in the whole space or in halfspaces bounded just from one-side. Polynomial Liouville properties and a Liouville theorem “at $ t = \infty$” are also presented.


CoffeeBreak


11:10-11:55K. Nyström (Uppsala Universitet)

Title: (Weighted) parabolic operators: fractional powers and the Kato square root problem

Abstract: In this talk I will discuss some recent results concerning second order parabolic operators with complex coefficients and fractional powers thereof. This leads us to study  weighted equations and the Kato square root problem for weighted parabolic operators.

12:00-12:45E. Lindgren (Uppsala Universitet)

Title: A doubly nonlinear evolutionary PDE related to Poincaré-type inequalities

Abstract: In this talk, I will discuss a doubly nonlinear evolutionary PDE that has a connection to extremals of Poincaré-type inequalities in Sobolev spaces. Special attention will be given to the large time behavior of solutions and the regularity of solutions. If time permits, I will also discuss some related equations with similar behavior.


Lunch

Thursday, June 16th

9:00-9:45N. Garofalo (Università di Padova)

Title: Two sub-Riemannian monotonicity formulas and some of their consequences

Abstract: I discuss two new monotonicity formulas for sub-Riemannian heat kernels and present some notable consequences of them. The content of my talk is joint work with Giulio Tralli.

9:50-10:35G. Tralli (Università di Padova)

Title: Mehler functions and CR-extension problems

Abstract: In this talk we will focus on the relationship between the hypoelliptic sub-Laplacian on Lie groups of Heisenberg type and some variants of the classical Ornstein-Uhlenbeck operator. In particular, we will discuss the explicit construction of the heat kernels for the relevant geometric extension problems starting from the knowledge of the Mehler kernels. As a main application, we will show how the properties of the Mehler-type functions provide integral representations and inversion formulas for the conformal fractional powers of the sub-Laplacian.
The talk is based on a joint project with N. Garofalo.



CoffeeBreak

11:10-11:55D. Pallara (Università del Salento)

Title: Mean value formulas for degenerate operators

Abstract: After recalling the classical mean value formulas for the Laplace operator and for uniformly parabolic operators with smooth coefficients, we deal with degenerate elliptic and parabolic operators with less regular coefficients. This case is discussed in the framework of the associated Lie groups, presenting the useful notions of geometric measure theory. We also show some applications to maximum principles and Harnack inequalities.

12:00-12:45G. Metafune (Università del Salento)

Title: $L^p$ estimates for degenerate problems in the half-space

Abstract: We study elliptic and parabolic problems governed by the singular elliptic operators
\begin{equation*}
\mathcal L =y^{\alpha_1}\Delta_{x} +y^{\alpha_2}\left(D_{yy}+\frac{c}{y}D_y -\frac{b}{y^2}\right),\qquad\alpha_1, \alpha_2 \in\mathbb R
\end{equation*}
in the half-space $\mathbb R^{N+1}_+=\{(x,y): x \in \mathbb R^N, y>0\}$.


Lunch

14:15-14:40J. Lenczewska (Wrocław University)

Title: Asymptotic expansion of the nonlocal heat content

Abstract: Let $(p_t)_{t \geq 0}$ be a convolution semigroup of probability measures on $\mathbb R^d$ defined by
\begin{equation*}
\int_{\mathbb R^d} e^{i\left<\xi,x\right>} p_t \, dx=e^{-t\psi(\xi)}\,, \quad \xi \in\mathbb R^d,
\end{equation*}
and let $\Omega$ be an open subset of $\mathbb{R}^d$ with finite Lebesgue measure. We consider the quantity
\begin{equation*}
H_{\Omega}(t)= \int_{\Omega}\int_{\Omega-x}p_t \, dy d x \: ,
\end{equation*}
which is called the heat content. We study its asymptotic expansion under mild assumptions on $\psi$, in particular in the case of the $\alpha$-stable semigroup.
This is a joint work with Tomasz Grzywny (WUST).

14:45-15:10A. Rutkowski (NTNU, Trondheim)

Title: Mean field game system and master equation associated with local and nonlocal diffusions in whole space

Abstract: We study well-posedness and regularity for the mean field game system in the whole Euclidean space, which consists of a backward Hamilton–Jacobi equation coupled with a forward Kolmogorov–Fokker–Planck equation. Both equations are driven by a Lévy operator which is allowed to have both local and nonlocal components. In addition to that, we derive the associated master equation and we prove that it has unique solution.
The talk is based on a joint work with Espen R. Jakobsen.

15:15-15:40K. Szczypkowski (Wrocław University)

Title: Relativistic stable operators with critical potentials

Abstract: We prove sharp local in time heat kernel estimates for the relativistic stable operators perturbed by critical potentials.
We discuss Hardy’s inequality and blow-up of solutions.


CoffeeBreak

16:00-16:25M. Litsgård (Uppsala Universitet)

Title: On the Dirichlet problem for Kolmogorov-Fokker-Planck type equations with rough coefficients

Abstract: In a recent joint work with Kaj Nyström existence and uniqueness results were established for weak solutions to Dirichlet problems in bounded and unbounded Lipschitz type cylinders for Kolmogorov-Fokker-Planck type equations with symmetric and uniformly elliptic coefficients. The proof is based on ideas by Brezis and Ekeland, that the solution can be obtained as the minimizer of a uniformly convex functional. I would like to put these results into some context, and briefly discuss the technique used in the proofs.

16:30-16:55T. Grzywny (Wrocław University of Science and Technology)

Title: Subordinated Markov processes: estimates for heat kernels and Green functions

Abstract: Let $(M, d)$ be a metric space and $\mu$ a Radon measure on $M$. Assume that $\{S(t)\}_{t\in T}$ is a Markov process on $M$ such that its transition function is absolutely continuous with $\mu$, where $T$ is a set of non-negative integers or a set of non-negative real numbers. By $A$ we denote the semigroup generator associated with the transition function of $\{S(t)\}$. For a Bernstein function $f$ we define a new semigroup with generator $-f(-A)$ that is a semigroup for the Markov process $\{S(K_t)\}$, where $\{K_t\}$ is a subordinator on $T$ associated with the function $f$. During the talk, there will be discussed estimates of the haet kernel/transition density and Green function of $\{S(K_t)\}$. The proofs are elementary and do not use estimates for transition probability of the subordinator.
The talk is based on joint work with Bartosz Trojan.

17:00-17:25C. Spina (Università del Salento)

Title: $L^p$ estimates for a class of degenerate operators

Abstract: We prove $L^p$-estimates for the operator
$$\mathcal L=\Delta_x+\Delta_y +c\frac{y}{|y|^2}\cdot\nabla_y-\frac{b}{|y|^{2}}=\Delta_x+L_y,$$
where $L_y=\Delta_y +c\frac{y}{|y|^2}\cdot\nabla_y-\frac{b}{|y|^{2}}$. The parameters $b,\ c$ are constant real coefficients subject to the condition $ D:=b+\left(\frac{M-2+c}{2}\right)^2> 0$.
We work in the space $L^p_c:=L^p(\mathbb{R}^{N+M}, |y|^c\, dxdy)$, motivated by the fact that the weight $|y|^c$ makes the operator symmetric in $L^2_c$ and we assume $M+c>0$, so that the measure $d\mu=|y|^c\, dx\, dy$ is locally finite on $\mathbb{R}^{N+M}$.
The operators $\Delta_x$, $L_y$ commute and the whole operator $\mathcal L$ satisfies the scaling property $I_s^{-1}\mathcal L I_s=s^2\mathcal L$, if $I_s u(x,y)=u(sx,sy)$. It is not difficult to see that $\mathcal L$ generates a semigroup in $L^p_c$ if and only if $L_y$ generates in $L^p(\mathbb{R}^M, |y|^c\, dy)$ and this is equivalent to $(M+c)\, \left|\frac{1}{2}-\frac 1 p\right|<1+\sqrt D$.
When $M=1$ and $b=0$, $L_y$ is a Bessel operator and both $\mathcal L=\Delta_x+B_y$ and $D_t-\mathcal L$ play a major role in the investigation of the fractional powers $(-\Delta_x)^s$ and $(D_t-\Delta_x)^s$, $s=(1-c)/2$, through the extension procedure” of Caffarelli and Silvestre [1]. For this reason, ${\mathcal L}$ and $D_t-\mathcal L$ are named the extension operators”.
When $M=1$, that is in the half-space $\mathbb{R}^{N+1}_+$, all the results of this paper, and much more, have been proved in [4] by taking advantage of sophisticated tools from operator valued harmonic analysis. More general, non symmetrizing weights $|y|^m\, dx\, dy$ are therein considered and both Dirichlet and Neumann boundary conditions. We refer the reader also to [2,3] for the case $b=0$ and with variable coefficients.
Here we use a different strategy and show that $L^p$-estimates for the pure $x$-derivatives, that is the boundedness of the operators $D_{x_ix_j}\mathcal L^{-1}$, follow from sub-solution estimates through an interpolation theorem in absence of kernels in homogeneous spaces due to Z. Shen. Sub-solution estimates, that is improving of integrability for (sub) solutions of the homogeneous equation $\mathcal Lu=0$, are proved by combining Cacciopoli estimates, weighted Sobolev embeddings and Moser iteration.

References

[1] L. Caffarelli,L. Silvestre: An extension problem related to the fractional Laplacean, Comm. Partial Differential Equations, 32 (2007), no. 7-9 1245-1260.
[2] H. Dong, T. Phan: On parabolic and elliptic equations with singular or degenerate coefficients, arxiv: 2007.04385 (2020)
[3] H. Dong, T. Phan: Weighted mixed-norm $L_p$ estimates for equations in non-divergence form with singular coefficients: the Dirichlet problem, arxiv: 2103.08033 (2021)
[4] G. Metafune, L. Negro, C. Spina: $L^p$ estimates for the Caffarelli-Silvestre extension operators, Journal of Differential Equations, Volume 316, (2022), Pages 290-345.

Friday, June 17th

9:00-9:45X. Zhang (Wuhan University)
online talk (Google Meet link)

Title : Second order McKean-Vlasov SDEs and kinetic Fokker-Planck-Kolmogorov equations

Abstract: In this work we study second order stochastic differential equations with measurable and density-distribution dependent coefficients.
Through establishing a maximum principle for kinetic Fokker-Planck-Kolmogorov equations with distribution-valued inhomogeneous term, we show the existence of weak solutions under mild assumptions. Moreover, by using the Hölder regularity estimate obtained recently in [1], we also show the well-posedness of generalized martingale problems when diffusion coefficients only depend on the position variable (not necessarily continuous). 
Even in the non density-distribution dependent case, it seems that this is the first result about the well-posedness of SDEs with measurable diffusion coefficients.

References

[1] F. Golse, C. Imbert, C. Mouhot, and A.s Vasseur, Harnack inequality for kinetic FokkerPlanck equations with rough coefficients and application to the Landau equation. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze

9:50-10:35E. Priola (Università di Pavia)

Title: Poisson process and sharp constants in $L^p$ estimates for a class of degenerate Kolmogorov operators

Abstract: This is a joint work with. S. Menozzi (Evry) and L. Marino (IMPAN). We  consider a possibly degenerate Kolmogorov-Ornstein-Uhlenbeck operator of the form $L={\rm Tr}(BD^2)+\langle Az,D\rangle $, where  $A$, $B $ are $N\times N $ matrices, $z \in \mathbb R^N$, $N\geq 1 $, which satisfy the Kalman condition which is equivalent to the  hypoellipticity condition. We prove the following stability result:  the Schauder and Sobolev estimates associated with the corresponding parabolic Cauchy problem remain valid, with the same constant, for the parabolic Cauchy problem associated with a second order perturbation of $L$, namely for $L+{\rm Tr}(S(t) D^2) $  where  $S(t)$ is a non-negative  $N\times N $ matrix depending continuously on $t \geq 0$. In the talk I  will concentrate on Sobolev estimates.   Our approach relies on the perturbative technique based on the Poisson process introduced in [Krylov-P.17].


CoffeeBreak
11:00-11:25L. Marino (Polish Academy of Sciences)

Title: Anomalous diffusion limit for a kinetic equation with a thermostatted interface

Abstract: We study the long time behavior of the solutions to a linear phonon Boltzmann equation in one spatial dimension, subject to a random mechanism of transmission, reflection or absorption at a point interface. Such a model naturally arises as the kinetic limit for a one-dimensional chain of harmonic oscillators with a random scattering of velocities, when one oscillator is in contact with a heat bath. Assuming a fast enough degeneracy of the scattering kernel and a slow (logarithmic) decay of the probability of absorption for low frequency phonons, we show that the solutions to our interface model exhibit a super-diffusive behavior in the long time limit, with a scaling parameter depending on the interplay between the decay velocities of the the scattering kernel and the drift. We also characterise such a limit as the unique weak solution of a fractional in space heat equation, with reflection-transmission-absorption at the interface. In a more probabilistic sense, the limit solution also corresponds to a certain stable process that is either transmitted, reflected or absorbed upon crossing the interface. Our proof relies on a merger between probabilistic techniques, exploiting that the kinetic dynamics away from the interface is indeed the Kolmogorov equation for a classic jump process, and some new analytic results on Dirichlet forms.
This talk is based on a work in progress in collaboration with T. Komorowski (Polish Academy of Sciences) and K. Bogdan (Wroclaw University of Science and Technology).

11:30-11:55G. Lucertini (Università di Bologna)

Title: Optimal regularity for degenerate Kolmogorov equations with rough coefficients

Abstract: We consider a class of degenerate parabolic equations satisfying a weak Hörmander condition, with coefficients that are measurable in time and Hölder continuous in the space variables. We prove the existence, estimates, and maximal regularity for the fundamental solutions of these operators. In order to do that, we adapt the classical Levi’s parametrix method. These operators can be viewed as the backward Kolmogorov operators associated to a family of SDEs and see applications in kinetic theory and mathematical finance. The regularity of the fundamental solution can be used to study the existence and the uniqueness for the solution of the SDE.

12:00-12:25A. Pesce (Università di Bologna)

Title: Density and gradient estimates for Kinetic SDEs with low regularity coefficients

Abstract: We discuss some recent results on Kinetic degenerate Kolmogorov SDEs with non-linear drift: we show two sided bounds and pointwise controls of its derivatives, under somehow minimal assumptions that guarantee that the equation is weakly well posed. 
These estimates reflect the transport of the initial condition by the unbounded drift through an auxiliary, possibly regularized flow.

12:30-13:15S. Pagliarani (Università di Bologna)

Title: A Yosida’s parametrix approach to Varadhan’s estimates for a degenerate diffusion under the weak Hörmander condition

Abstract: We adapt and extend Yosida’s parametrix method, originally introduced for the construction of the fundamental solution to a parabolic operator on a Riemannian manifold, to derive Varadhan-type asymptotic estimates for the transition density of a degenerate diffusion under the weak Hörmander condition. This diffusion process, widely studied by Yor in a series of papers, finds direct application in the study of a class of path-dependent financial derivatives known as Asian options. We obtain a Varadhan formula for the logarithm of the transition density in terms of the optimal cost function of a deterministic control problem associated to the diffusion. We also derive an asymptotic expansion of the cost function, expressed in terms of elementary functions, which is useful in order to design efficient approximation formulas for the transition density.


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